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# Triangle, trisected

### #1

Posted 11 March 2013 - 09:08 AM

**Warm-up problem:**

Bisect the angles of a triangle.

Describe the point(s) where each bisector first intersects one of the others.

**Now try this:**

Trisect the angles of a triangle.

Describe the points where each trisector first intersects one of the others.

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #2

Posted 11 March 2013 - 12:38 PM

Do we have to name that point?

### #3

Posted 11 March 2013 - 03:19 PM

Do we have to name that point?

Spoiler for

You're right about the bisector case.

But for the trisector case there is more than one point.

In fact there are three places where a trisector of one angle **first** intersects a trisector of one of the other angles.

And there is something special about those three points.

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #4

Posted 12 March 2013 - 09:49 AM

Do we have to name that point?

Spoiler for

You're right about the bisector case.

But for the trisector case there is more than one point.

In fact there are three places where a trisector of one angle

firstintersects a trisector of one of the other angles.And there is something special about those three points.

### #5

Posted 14 March 2013 - 04:38 PM

Do we have to name that point?

Spoiler for

You're right about the bisector case.

But for the trisector case there is more than one point.

In fact there are three places where a trisector of one angle

firstintersects a trisector of one of the other angles.And there is something special about those three points.

Spoiler for i think

Actually they are not collinear.

That being the case, they form a triangle.

So the OP really asks: what is special about the triangle formed by these three points?

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #6

Posted 14 March 2013 - 09:47 PM Best Answer

### #7

Posted 15 March 2013 - 05:25 AM

Spoiler for It seems...

Correct. Good job.

I wonder if there is a proof of this that is not overly complex?

Edit:

Well, No. I just found the proof, and it's not beautiful for its simplicity.

You start with the Law of Sines, and 2 1/2 pages later you have a symmetrical expression for one side.

"Do not try this at home."

**Edited by bonanova, 15 March 2013 - 06:23 AM.**

Comment on proof

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #8

Posted 18 March 2013 - 01:01 PM

Spoiler for It seems...Correct. Good job.

I wonder if there is a proof of this that is not overly complex?

Edit:

Well, No. I just found the proof, and it's not beautiful for its simplicity.

You start with the Law of Sines, and 2 1/2 pages later you have a symmetrical expression for one side.

"Do not try this at home."

oh i am so sorry .....i mistook it for the eulers line....my bad...

### #9

Posted 24 March 2013 - 03:09 AM

COMBINATION: 7129

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